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ABSTRACT
The effect of an external axial magnetic field on the liquid metal flow produced by co-rotation of the top and bottom disks in a vertical cylindrical container with a vertical temperature gradient is numerically analyzed. The governing Navier–Stokes, energy, and potential equations along with appropriate boundary conditions are solved using the finite-volume method. Comparisons with the previous results were performed and found to be in excellent agreement. It was observed that the Reynolds number is increased, and the axisymmetric basic state loses stability for circular patterns of axisymmetric vortices and spiral waves. In the mixed convection case the axisymmetric mode disappears, giving an asymmetric mode m = 1. It was also found that the primary thresholds, Recr corresponding to modes m = 1 and 2, increase with an increase in Hartmann number (Ha). We can therefore conclude that when the magnitude of the magnetic field exceeds a certain value, the instability becomes a steady bifurcation. Finally, stability diagrams were established according to the numerical results of this investigation. These diagrams show the evolution of primary thresholds as a function of Hartmann number for various values of Richardson number.
Nomenclature
| B | = | magnetic field, Tesla |
| FLr | = | dimensionless radial Lorentz force |
| FLz | = | dimensionless axial Lorentz force |
| FLθ | = | dimensionless azimuthal Lorentz force |
| g | = | acceleration of gravity, m/s2 |
| H | = | height of the cylinder, m |
| Ha | = | Hartmann number |
| J | = | dimensionless current density |
| N | = | interaction parameter, (=Ha2/Re) |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number (=ν/α) |
| R | = | radius of the cylinder, m |
| r | = | dimensionless radial coordinate |
| z | = | dimensionless axial coordinate |
| Re | = | Reynolds number (=ΩR2/ν) |
| Ri | = | Richardson number (=g β(Th − Tc)/Ω2R) |
| T | = | temperature, K |
| u | = | dimensionless radial velocity |
| v | = | dimensionless axial velocity |
| w | = | dimensionless azimuthal velocity |
| α | = | thermal diffusivity of the fluid, m2 s−1 |
| β | = | thermal expansion coefficient, K−1 |
| γ | = | aspect ratio (= H/R) |
| Θ | = | dimensionless temperature, [= (T − Tc)/(Th − Tc)] |
| ν | = | kinematic viscosity of the fluid, m2/s |
| ρ | = | density of the fluid, kg/m3 |
| σ | = | electrical conductivity, Ω−1 · m−1 |
| τ | = | dimensionless time |
| Φ | = | dimensionless electric potential |
| Ω | = | angular velocity, rad s−1 |
| Subscripts | = | |
| cr | = | critical value |
| c | = | cold |
| h | = | hot |
| r, z, θ | = | radial, axial, and azimuthal directions, respectively |
Nomenclature
| B | = | magnetic field, Tesla |
| FLr | = | dimensionless radial Lorentz force |
| FLz | = | dimensionless axial Lorentz force |
| FLθ | = | dimensionless azimuthal Lorentz force |
| g | = | acceleration of gravity, m/s2 |
| H | = | height of the cylinder, m |
| Ha | = | Hartmann number |
| J | = | dimensionless current density |
| N | = | interaction parameter, (=Ha2/Re) |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number (=ν/α) |
| R | = | radius of the cylinder, m |
| r | = | dimensionless radial coordinate |
| z | = | dimensionless axial coordinate |
| Re | = | Reynolds number (=ΩR2/ν) |
| Ri | = | Richardson number (=g β(Th − Tc)/Ω2R) |
| T | = | temperature, K |
| u | = | dimensionless radial velocity |
| v | = | dimensionless axial velocity |
| w | = | dimensionless azimuthal velocity |
| α | = | thermal diffusivity of the fluid, m2 s−1 |
| β | = | thermal expansion coefficient, K−1 |
| γ | = | aspect ratio (= H/R) |
| Θ | = | dimensionless temperature, [= (T − Tc)/(Th − Tc)] |
| ν | = | kinematic viscosity of the fluid, m2/s |
| ρ | = | density of the fluid, kg/m3 |
| σ | = | electrical conductivity, Ω−1 · m−1 |
| τ | = | dimensionless time |
| Φ | = | dimensionless electric potential |
| Ω | = | angular velocity, rad s−1 |
| Subscripts | = | |
| cr | = | critical value |
| c | = | cold |
| h | = | hot |
| r, z, θ | = | radial, axial, and azimuthal directions, respectively |